Riemann Roch: structure of genus 1 curves

This talk is about the Riemann Roch theorem in the spacial case of genus 1 curves or Riemann surface. We show that a compact Riemann surface
satisfying the Riemann Roch theorem for g=1 is isomorphic to a nonsingular plane cubic. We show that this is topologically a torus, and use this to show that it can be represented as C/L for a lattice.

This video is based on a discussion section of an online algebraic geometry course.

[hausdorffm]

2:15 For genus-1 curve, l(D) = deg D if D >0.
In fact, by deg K = deg(dz) = 0, if a function f with (f) + K -D >= 0, then 0=deg(f)>deg(-K+D) = deg D >0 and this contradicts. So, l(K-D) =0. Thus, l(D) = deg D for D >0 on genus-1.

13:52
The definition of dx/y depends on the local coordinates, is it well-defined on the plane curve? where is it defined on?

[tomasmanriquezvalenzuela]

Do you recomend any book to learn about algebraic geometry?

[rorycannon]

Hey, mid-college level student here really interested in math like this but unsure where to start. Can someone point me to a video so i can just go up watching in a series?